This is Chapter 3 – Functions for the Year 12 Specialist Mathematics course.

This chapter extends functional reasoning beyond the techniques used in Mathematical Methods and asks students to analyse how more complex functions are constructed, restricted and transformed. A strong understanding of structure is essential, since later Specialist Mathematics topics rely on recognising how algebraic changes affect graphical behaviour.

We begin with composite functions, where one function is applied within another. Students must substitute carefully and consider the domain of the resulting composition rather than focusing only on the algebra. Inverse functions then require students to determine when a function is one-to-one, apply suitable domain restrictions and interpret the symmetry between a function and its inverse.

Reciprocal functions introduce discontinuities and asymptotic behaviour. Students examine how zeros, sign changes and end behaviour in an original function influence the graph of its reciprocal, strengthening the connection between algebra and graph analysis.

Rational functions extend these ideas further, often combining factorisation, domain restrictions and asymptotes within the same problem. Absolute value functions introduce piecewise behaviour and require students to reason carefully about how different parts of a graph are reflected or transformed. Sketching and algebraic analysis are used together, with students expected to justify features such as excluded values and asymptotes from the equation rather than relying only on technology.

These function skills support later calculus, parametric modelling and vector calculus, where domains, inverses and transformations frequently affect the validity of a solution. Functional analysis is also important in mathematical modelling, computer science and engineering, where complex systems are often built by combining simpler relationships.

Subchapters:

• 3A: Composite functions
  
• 3B: Inverse functions
• 3C: Reciprocal functions
• 3D: The reciprocals of other functions
• 3E: Rational functions
• 3F: Absolute value functions

By the end of this chapter, students will be able to analyse composite, inverse, reciprocal, rational and absolute value functions with careful attention to domains, restrictions and graphical structure.